Detempering: Difference between revisions

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Detempering is one way among many to create a [[neji]], or a JI scale approximating a given scale.

== One-to-one detemperings ==

The following ~~is one possible definition~~ for one-to-one detemperings: A JI scale ''S'' is a ''one-to-one detempering'' if on the [[JI subgroup]] <math>A \leq \mathbb{Q}_{>0}</math> generated by the intervals of ''S'', there exists a [[val]] ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ. ~~Equivalently, it is a detempering of an [[equal temperament]] under some mapping where each note of the equal temperament is matched to exactly one JI note which tempers to the note.~~ The two definitions are equivalent because if a detempering of an n-note equal temperament ''v'' is one-to-one, then the first definition follows by the additivity of ''v'', and given the first definition the injectivity is immediate.

The following are two equivalent definitions for one-to-one detemperings:

# A JI scale is a ''one-to-one detempering'' if it is a detempering of an [[equal temperament]] under some mapping where each note of the equal temperament is matched to exactly one JI note which tempers to the note.

# A JI scale ''S'' is a ''one-to-one detempering'' if on the [[JI subgroup]] <math>A \leq \mathbb{Q}_{>0}</math> generated by the intervals of ''S'', there exists a [[val]] ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ.

The two definitions are equivalent because if a detempering of an n-note equal temperament ''v'' is one-to-one, then the first definition follows by the additivity of ''v'', and given the first definition the injectivity is immediate.

As suggested by its name, the property is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is a CS but not that it is a one-to-one detempering, there is a unique set map <math>v : \{\text{intervals of $S$}\} \to \mathbb{Z}</math> that witnesses that ''S'' is a CS and satisfies ''v''(''S''[''i'']) = ''i'' for all ''i''. Thus a CS scale ''S'' is a one-to-one detempering if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.

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 Detempering is one way among many to create a [[neji]], or a JI scale approximating a given scale.
 == One-to-one detemperings ==
-The following is one possible definition for one-to-one detemperings: A JI scale ''S'' is a ''one-to-one detempering'' if on the [[JI subgroup]] <math>A \leq \mathbb{Q}_{>0}</math> generated by the intervals of ''S'', there exists a [[val]] ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ. Equivalently, it is a detempering of an [[equal temperament]] under some mapping where each note of the equal temperament is matched to exactly one JI note which tempers to the note. The two definitions are equivalent because if a detempering of an n-note equal temperament ''v'' is one-to-one, then the first definition follows by the additivity of ''v'', and given the first definition the injectivity is immediate.
+The following are two equivalent definitions for one-to-one detemperings:
+# A JI scale is a ''one-to-one detempering'' if it is a detempering of an [[equal temperament]] under some mapping where each note of the equal temperament is matched to exactly one JI note which tempers to the note.
+# A JI scale ''S'' is a ''one-to-one detempering'' if on the [[JI subgroup]] <math>A \leq \mathbb{Q}_{>0}</math> generated by the intervals of ''S'', there exists a [[val]] ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ.
+The two definitions are equivalent because if a detempering of an n-note equal temperament ''v'' is one-to-one, then the first definition follows by the additivity of ''v'', and given the first definition the injectivity is immediate.
 As suggested by its name, the property is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is a CS but not that it is a one-to-one detempering, there is a unique set map <math>v : \{\text{intervals of $S$}\} \to \mathbb{Z}</math> that witnesses that ''S'' is a CS and satisfies ''v''(''S''[''i'']) = ''i'' for all ''i''. Thus a CS scale ''S'' is a one-to-one detempering if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.